# U = √(GM/r)

Artificial satellites are rarely placed in retrograde orbit. This is partly due to the extra velocity required to go against the direction of the rotation of the earth.

So, the required speed for rotating without falling, for a geostatic and one moving at the opposite direction of a geostatic, is not the same and equal to u=√(GM/r)?

For Polar Orbits it's u=√(GM/r)?

Related Astronomy and Astrophysics News on Phys.org
Janus
Staff Emeritus
Gold Member

Artificial satellites are rarely placed in retrograde orbit. This is partly due to the extra velocity required to go against the direction of the rotation of the earth.

So, the required speed for rotating without falling, for both a geostatic and one moving at the opposite direction of a geostatic is not u=√(GM/r)?

For Polar Orbits it's u=√(GM/r)?
The speed is the same, but that speed is relative to a non-rotating frame of reference, not the surface of the Earth. Since the satellites are launch from the moving surface, if you launch in the same direction that the Earth is rotating, you have a "running start" as it were as you alrady have some of the speed that you need to achieve orbit. If you launch the other way, you first have to negate the speed you have in the wrong direction and then make up for the speed advantage you had when launching with the rotation.

D H
Staff Emeritus

Wikipedia said:
Artificial satellites are rarely placed in retrograde orbit.[citation needed]
Citation needed, indeed. The article does mention sun synchronous orbits, and those orbits are (slightly) retrograde. The vast majority of the Earth observing satellites in low Earth orbit are in sun-synchronous orbits.

So, the required speed for rotating without falling, for a geostatic and one moving at the opposite direction of a geostatic, is not the same and equal to u=√(GM/r)?
While orbital velocity does not depend on orbital inclination, the amount of energy needed to launch a satellite into orbit depends on orbital inclination and on the latitude of the launch site.

So, wikipedia implies that the reason retrograte sattelites are avoided, is that more boosting energy is required for the launch? This is weird, why is that such a big problem?

Jonathan Scott
Gold Member

So, wikipedia implies that the reason retrograte sattelites are avoided, is that more boosting energy is required for the launch? This is weird, why is that such a big problem?
Why would anyone (apart from Israel with its specific geographic requirements) want to launch them in the retrograde direction when the forward direction is significantly cheaper?

Near the equator, the earth's speed of rotation is around 25,000 miles per 24 hours, which is of the order of 1000mph. A retrograde orbit therefore requires an additional 2000mph, so the speed needs to be about 12% faster and the kinetic energy required is about 25% more.

Do geostatic sattellites need constant boosting or they can continue orbiting forever without boosting?

Jonathan Scott
Gold Member

Do geostatic sattellites need constant boosting or they can continue orbiting forever without boosting?
Geostationary satellites orbit in free fall orbit at a distance calculated to ensure that the orbit takes one day, so they go round the earth in the same time as the earth rotates and hence appear to be above the same spot on the equator all the time. They do not need boosting.

(Technically, the orbit is not exactly 24 hours but rather a "sidereal day" which is about 4 minutes less, because we measure a normal day from when the sun appears to be in the same position in the sky, but since the Earth is moving around the sun, that direction changes from day to day, completing one rotation per year. If we measure the Earth's rotation against the stars instead, it takes about 23 hours 56 minutes to complete a rotation, so that is also the orbital period needed for a geostationary satellite).

Are you 100% certain that they don't need boosting?

By boosting I mean that some artificial engine must add momentum to it, otherwise it will lose its speed.

Jonathan Scott
Gold Member

Are you 100% certain that they don't need boosting?

By boosting I mean that some artificial engine must add momentum to it, otherwise it will lose its speed.
Definitely no boosting required.

At that altitude, there's nothing much to cause it to lose its speed. Objects in low earth orbits (say 100 miles to 1000 miles) tend to decay relatively quickly (over months or years) because of friction with tiny traces of atmosphere, but geostationary orbit is much higher up (about 22,000 miles).

A geostationary satellite will however need the occasional tweak to put it back in the right place, as there are tiny tidal and precession effects (caused mainly by the fact that the earth isn't a uniform perfect sphere) which eventually cause it to wander, so it does need a supply of fuel for thrusters which will last for its active lifetime.

http://en.wikipedia.org/wiki/Geostationary_orbit" [Broken]

Last edited by a moderator:

A retrograte one at the geostationary height needs no boosting either?

Jonathan Scott
Gold Member

A retrograte one at the geostationary height needs no boosting either?
The formula in the title of this thread applies to all circular orbits in any direction (polar, equatorial or anywhere in between) as seen from a non-rotating frame of reference at rest with respect to the earth (or the gravitational source in general). Satellites at the same height r from the center of the earth (which has mass M), moving in any horizontal direction with the speed u as given by this formula will be in a circular orbit.

However, as the earth is rotating, the speed relative to the surface of the earth is not simply u but also needs to take into account the motion of the earth.

A retrograde satellite at exactly the geostationary height would still go round the earth once a day in a circular orbit, but as the earth turns the other way it would appear to go round twice a day as seen from the ground. It would also be very unpopular, as there would be a significant risk of it crashing into a geostationary satellite at nearly 14,000mph.